Hey, guys. So for this video, we're going to talk about a concept called the self-inductance. Now if you've seen our video on mutual inductance, this is going to be very, very similar to that discussion with a few small differences. And in case you haven't seen that video yet, pay attention because it's going to come up later. Let's check it out. So the whole idea of self-inductance and the whole thing here is that a current-carrying wire has the ability to induce an EMF on itself by changing its magnetic flux. Flux. So let's take a look at how that happens. Here I have a single coil of wire, and it turns into a loop like this. Right? So it turns into sort of like a solenoid kind of thing. So if this solenoid-looking thing has a current going through it, it generates a magnetic field. And if this magnetic field is passing through a surface, then that means that we have a magnetic flux. That flux is just ba ∙ cosθ . What happens is that our magnetic field points off to the right and our area vector is going to be perpendicular to that surface, so that also points to the right like this. And because those two things point along the same exact direction, that cosθ term just goes to 1. Alright. So what happens is we have the flux that depends on the current or sorry, we have the flux that depends on this magnetic field, but what happens is that's only just for one of the turns. If we wanted to figure out what the total amount of flux is, that would depend on n, which is the number of turns, and the magnetic field and the area. The problem is that this magnetic field also depends on another variable. Remember that this magnetic field depends on the amount of current that passes through it. So B depends on I. So let's see what's happening here. We have the flux that depends on the magnetic field, but the magnetic field depends on the current. So that means that there is a relationship between the flux and the current as well. So basically what I'm just saying is that there is a proportionality. So we see that φB is proportional to the currents. And all I'm saying here is that there is sort of like a mathematical if you were to sort of write out an equation for this, this just turns out to be n on the left. Right? So that's the total amount of flux is equal to I, but the proportionality of the constant that goes out here is called L. And it's called the, it's called the self-inductance. And basically what the self-inductance represents is it's the ability for a current-carrying wire to generate or induce an emf on itself by changing its magnetic flux. That's really what it just represents. And the equation for that is pretty straightforward. We can actually get it from this, formula right here. So it's just going to be n ∙ βB ÷ I. And the units for that are given as Henrys. So if you've seen the video on mutual inductance, again, it's going to be the same exact thing. And in terms of more fundamental units, that's actually going to be a Weber per ampere. So we have flux in Webers, and we have amperes and currents on the bottom. Okay? So the next really important thing that you need to know is that it depends only on the number of turns, which is n, and it depends on the shape of the coil, which is going to give us that flux value. So what happens is we're going to see that this current value is always going to cancel out. So even though we're going to use this equation to calculate the self-inductance, we're going to see that the current will cancel out, and this L is only sort of like a physical property of the coil. Now before we get into it an example, the last thing I want to point out is that now we actually have a different formula we can use for the self-induced EMF. Emf. So we know that Faraday's law tells us that we can relate an emf with the number of turns and the change in the magnetic flux over the change in time. And we are perfectly okay to use that equation. But we can also write the self-induced EMF in terms of this self-inductance term that we just found out, and that's just going to be negative L ∙ ΔIΔt. And just in case you need to know where this equation comes from, it actually just comes from this relationship right here. So if we were to just divide both sides by delta t, so if we wanted to figure out how each one of these things was changing with time, remember that n ∙ ΔβBΔt is the definition of what EMF is. And so both of these things, end up being equal to each other. So we just have a different expression that we can use for this EMF. Right? So if you have the change in flux over change in time, you can figure out what the EMF is. But now using this self-inductance, if you have the change in currents over change in time, you could also figure out what the EMF is. Alright? So that's basically it. Let's go ahead and check out an example. In this example, we're going to be calculating what the expression is for the self-inductance of a single current-carrying loop of wire. So if we want the self-inductance, remember that is going to be L. So L is our target variable here, and now we have 2 equations that involve L. 1 of them involves L and the self-induced EMF. That's going to be negative L ∙ ΔIΔt. But the thing is is that I don't have any information about the self EMF, and I have no information about how the current is changing over time. So this is not the equation I'm going to use. Instead, the other equation is that L = n ∙ βB ÷ I. Alright? So let's go ahead and work with this. We have a single current-carrying loop of wire. So if it's a single loop, that just means that n = 1. So that means that sort of this just becomes 1 right here. And so L = βB ÷ I. So let's take a look at here. Right? So we have a loop of wire that has a current that's going through it. So using our right-hand rule, if you were to take your right fingers and curl them in the direction of this, your thumb should be pointing into the page. So that means that the magnetic field points in this direction. And if it's a loop of wire like this, then that means that the area vector also points in the same direction, sort of into the page. So that means that both of these things, because they sort of point along the same direction away from you into the page, then that means that this flux right here, this φB is equal to BA ∙ cosθ. But remember, because those things point in the same direction, this is just going to be 1. So that means the flux is just the magnetic field times the area. But for a loop of wire, we can actually figure out what that magnetic field is. So let's go ahead and write out what those equations are. So we have so Bloop × A. So remember that Bloop is going to be μ0 I ÷ 2 × little r where that's the radius. And then the area, the cross-sectional area right here is just the area of the circle, which is going to be π × r2. So we can just do some canceling out right here. We have an r2 on top and an r on the bottom, so we just cancel that out and leave 1 R remaining. And that means that our flux here, φB, is just equal to μ0 I × π × r ÷ 2. So now that we have an expression for the flux, in order to find out what the self-inductance is, we just need to plug this formula back inside of this. So that means that our self-inductance L is just going to be μ0 I × π × R ÷ 2, that's the self that's the magnetic flux that we just found, divided by the currents. So what we see here is that the currents will cancel out from the top and the bottom exactly like we said it would. And so this L basically just is μ0 I π oh, sorry. There's no I anymore. So it's just μ0 × π r ÷ 2. So it only basically just represents or it's only dependent on the number of coils or in the number of turns, which in this case was 1, and the shape of the coil itself. So we have that π × the r. It sort of is from, like, the circular shape of the loop. Okay? So this is the self-inductance for a coil of wire, and it's sort of like a physical property that just depends on that shape of the coil. Alright, guys. That's it for this one. Let me know if you have any questions.
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30. Induction and Inductance
Self Inductance
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